There are many small local medical lab clinics around Taipei which can carry out blood and other tests, often including X-rays etc. Look in Google for “檢驗所”. Prices depend on what you tests you want. Many will offer packages of tests covering most important areas for $2000-$6000. They will give you the results and usually it includes a guide to what are normal ranges. You would probably need to take this to a GP to get more detailed feedback about how to deal with anything that is out of range.
Another, more expensive option is to get a full body health check done at a specialist centre. One of the best and pretty foreigner friendly is Lian An:
https://www.lianan.com.tw/en/
(They also have health checking services for employees, but that would depend on your company having an arrangement with them.)
For private customers, they have a range of packages from $22k to $118k. You’re unlikely to need the most expensive one, as this is basically a VIP service which has a dedicated staff member assigned to go around with you all day to each testing station. The lower priced packages, in particular A, B or C are still very comprehensive and cover pretty much all the important stuff, including chest X-rays, ECG, ultrasound scans etc (which package you want will depend on your age and family background…e.g. a colonoscopy is probably a good idea include if you’re over around 50 or earlier if there is history of colon problems in your family).
The testing day (or half day if you aren’t having some of the more complicated/lengthy tests) is well managed and they look after you throughout, moving you to each test station one after another, so the time passes quite fast. They assign a main (shared) assistant (English speaking if needed) who will keep an eye on you and guide you through the process (sometimes someone else will take over if your main contact is busy, but she will always reappear and check things are OK).
At the end of the testing day they already have most of the results ready (there is a big testing lab onsite), and you have a consultation with a doctor who takes time to go through everything and answer your questions. He will suggest any areas you should follow up with a specialist. About 2-3 weeks later you get a hardcopy report which gives all the results in detail (in english and chinese) and again makes recommendations of what you should to improve your health or if follow-up checks should be done at a main hospital.
Some of the bigger hospitals offer similar dedicated health-check services, though when I was looking around, I found they are primarily aimed at local users, and while they can be cheaper, there seemed to be little language help available. So for example, they may only have a list of the tests included in chinese. Great if you have strong language skills, but on health matters I didn’t want to risk not understanding what I’m being told or getting things poorly explained in broken english.